Question

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). $$ a_{1}=2 \text { and } a_{n+1}=2 a_{n} \text { for } n \geq 1 $$

Short answer

The function is \(f(n) = 2^n\).

Step-by-step solution

Understand the Problem

The given sequence starts with \(a_1 = 2\), and each subsequent term is twice the previous term. This recursive formula can be expressed as \(a_{n+1} = 2a_n\) with \(a_1 = 2\). Our task is to find a closed formula \(f(n)\) such that \(a_n = f(n)\).

Observe the Pattern

Let's manually calculate a few terms of the sequence to observe a pattern, using the recursion: - \(a_1 = 2\) (given)- \(a_2 = 2 \times a_1 = 2 \times 2 = 4\)- \(a_3 = 2 \times a_2 = 2 \times 4 = 8\)- \(a_4 = 2 \times a_3 = 2 \times 8 = 16\)Notice that \(a_n\) appears to be powers of 2.

Identify a Pattern in Terms

From Step 2, the calculated terms are:- \(a_1 = 2^1\)- \(a_2 = 2^2\)- \(a_3 = 2^3\)- \(a_4 = 2^4\)It seems that \(a_n = 2^n\). The pattern is that each term is \(2\) raised to the power of its position number \(n\).

Generalize the Formula

Based on the pattern observed, we hypothesize that the closed-form solution for \(a_n\) is \(f(n) = 2^n\). This formula should work for any \(n\geq1\), satisfying the recursive definition and initial condition.

Verify the Formula Fits the Recursion

To verify, check if \(f(n)\) satisfies the recurrence.\[a_{n+1} = 2a_n \f(n+1) = 2^{n+1} = 2 \times 2^n = 2a_n \\]Since \(a_{n+1} = 2a_n\) holds true under the formula \(f(n) = 2^n\), our formula is consistent with the recursion.

Confirm Initial Condition

Finally, double-check the initial condition is met:For \(n = 1\), \(f(1) = 2^1 = 2\), which matches the provided initial condition \(a_1 = 2\). Thus, \(f(n) = 2^n\) satisfies all requirements.

Key concepts

Closed-form Solution

A closed-form solution is an explicit formula that allows you to calculate terms in a sequence directly, without needing to rely on the previous terms. In the context of a recursive sequence, a closed-form solution simplifies the complex process of iteratively applying rules to generate sequence terms. It often involves recognizing a pattern that emerges from the recursive process and expressing it as a single mathematical expression.
In our exercise, we started with the recursive relation \(a_{n+1} = 2a_n\), with the initial value \(a_1 = 2\). By observing the first few terms \(a_2 = 4\), \(a_3 = 8\), and so on, we identified a pattern. This pattern led us to propose that each term \(a_n\) is a power of 2, specifically \(2^n\).
This formula, \(f(n) = 2^n\), serves as our closed-form solution. It allows us to find any term in the sequence without recursing back through all previous terms. Providing closed-form solutions is vital in mathematical fields, as it simplifies calculation and helps in understanding the behavior and properties of sequences more generally.

Recurrence Relation

A recurrence relation is a formula that defines each term of a sequence using one or more of its preceding terms. These relations are foundational in sequences and series, enabling the description of patterns where each element depends on the earlier ones. They are crucial for developing closed-form solutions and understanding how sequences evolve.
In our exercise, the recurrence relation is given by \(a_{n+1} = 2a_n\). This implies that each term after the first is twice the previous term, forming a simple yet powerful geometric progression. By iterating this relation from the initial condition, you can build the sequence step-by-step, which is essential for recognizing the pattern leading to the closed-form solution.

• This method is particularly useful when:
  • Building sequences iteratively
  • Verifying closed-form solutions
  • Analyzing the dynamic behavior of sequences

Understanding recurrence relations paves the way for exploring more complex patterns and developing robust analytical skills in mathematics.

Initial Condition

The initial condition in a recursive sequence provides the starting point from which all other terms can be derived. It is a crucial part of the definition of a sequence, ensuring that each term can be uniquely determined.
For our problem, the initial condition is \(a_1 = 2\). This first term anchors the sequence and is essential for applying the recurrence relation \(a_{n+1} = 2a_n\) correctly. Without an initial condition, a recurrence relation would not sufficiently define a unique sequence.
Initial conditions are vital because:

• They help establish the concrete starting point of the sequence
• Generate all subsequent terms accurately
• Are used to verify the closed-form solution matches the sequence's requirements

By confirming that our proposed formula \(f(n) = 2^n\) fits both the recurrence relation and the initial condition, we ensure it accurately represents the sequence for all \(n \geq 1\). Thus, the initial condition not only assists in computing terms but also in validating the approach and solution.